Strevens - Inferring Probabilities From Symmetries

8/12/2019 Strevens - Inferring Probabilities From Symmetries

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Inferring Probabilities From Symmetries

Michael Strevens

Nos, 32:231246, 1998

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This paper justies the inference of physical probabilities from symmetries.I supply some examples of important and correct inferences of this variety.Two explanations of such inferencesan explanation based on the princi-ple of indifference and a proposal due to Poincar and Reichenbachareconsidered and rejected. I conclude with my own account, in which theinferences in question are shown to be warranted a posteriori, provided thatthey are based on symmetries in the mechanisms of chance setups.

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This version of the paper has been updated to use the terminology andnotation of Bigger than Chaos (Harvard University Press, 2003). The gureshave been revised accordingly, and made somewhat prettier.

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Thanks to Barry Loewer and Tim Maudlin for helpful comments.

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Consider a perfectly symmetrical dodecahedron (gure 1). Paint one facered, and it may be used as a die. What is the probability that such a die, whentossed, lands red face up? Without having touched, or even laid eyes on, suchan object, it is possible to give a rm answer: the probability is one in twelve.Find such a die, and experience will bear out this assertion. The red face willland uppermost roughly one-twelfth of the time. We have made, apparently,an a priori inference to a fact about the world.

Figure 1: A dodecahedral die

How is it that we are able to infer the probability with such success?The conventional answer to this question is that we invoke a rule called theprinciple of indifference (or the principle of insufcient reason). 1 This rule

1. For early (eighteenth century) examples of the use of a principle of indifference tocalculate probabilities concerning celestial objects, see van Fraassen (1989), 2978. Laplaceis among those cited; however, in his more philosophical writings, he seems to have realizedthat, inasmuch as probabilities generated by indifference are determined by ignorance alone,they may not correspond to real physical probabilities. Exegesis is difcult because Laplacesdeterminism inclined him against the existence of physical probabilities, and also because

(perhaps for this very reason) he often failed to clearly distinguish physical and epistemicprobability (see Hacking (1975), 1313 for a discussion of the ways that Laplace did and didnot make this distinction).

More recently E. T. Jaynes (1983) has applied a principle of indifference to determineprobabilities in chance setups. In his most philosophical moments, Jaynes, like Laplace,admits that probabilities determined by ignorance are not necessarily the correct physical

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tells us that, in the absence of any known reason to assign two events differing

probabilities, they ought to be assigned the same probability. Applied to thedodecahedral die, the principle allows us to reason as follows. The die may land with any of twelve faces uppermost. We know nothing that distinguishesthese twelve possibilities, so we must assign each an equal probability. Sincethe sum of these probabilities must equal one (some face must be uppermost),each possibility should be assigned a probability of one-twelfth.

This explanation of our probabilistic prowess cannot be right. The princi-ple of indifference is not even the right kind of rule to explain our successfulinference. It purports to tell us which probabilities are rational given ourignorance, helping us to do the right thing under adverse epistemic circum-stances. But our achievement is not mere rationality; it is truth. We inferthe correct physical probability for the event in question. This suggests thatour inference is not based on ignorance at all, and thus not based on a prin-ciple of indifference or insufcient reason. Two arguments support thissuggestion.

First, it is surely the case that we can never reliably get from ignoranceto truth, because the nature of the world is independent of our epistemic

decits. The fact we do not know anything about A does not constrain theway things are with A.

Second, in the sort of case under discussion, we have an inferential dis-position that is incompatible with the claim that our inference is based onignorance. The disposition is as follows. Suppose that you are certain thatsome particular dodecahedral die is absolutely symmetrical, externally andinternally (an internally symmetrical die is not loaded). Suppose also that,on repeated throwing, the red face of this die lands uppermost one-half of the

probabilities. (They are correct only if the problem is well posed, but knowing that theproblem is well posed requires some kind of real knowledge about the situation.)

These admissions of the epistemological limitations of the indifference principle only make it the more puzzling that our symmetry-based estimates of probability are so oftencorrect.

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time. Should you revise your estimate of the probability? Not at all. Given

the symmetry of the die, the probability of a red face must be one-twelfth.That the frequency of red faces is one-half can only be an accident. Thisinferential dispositionour refusal to revise the probability in the light of the new datasuggests that our knowledge of the probability, far from beingbased on ignorance, is based on knowledge so certain that it overrides any amount of information about outcomes. In the case described, that can only be our certain knowledge of symmetries. 2

The knowledge from which we infer the probability, then, is our certainknowledge of the symmetry of the die. It is the purpose of this paper toexplain, and to justify, the rule of inference that takes us from knowledgeof physical symmetries to knowledge of actual physical probabilities. (Istress once more that this rule is very different from the principle of indiffer-ence, which takes us from symmetries in our knowledgeor more exactly,ignoranceto rational probabilities.) 3 I call the inference from symmetriesto probabilities non-enumerative statistical induction, or for short.4 Enu-merative statistical induction , by contrast, is that form of reasoning in whichwe infer probabilities from frequencies of outcomes, as when we conclude

that smoking causes cancer with a certain probability by perusing the relevantmedical statistics.

The paper has four parts. In the rst part I show that non-enumerativestatistical induction plays an important role in several sciences. Thus the

2. Of course, we are not normally absolutely certain about symmetries. A frequency of red faces signicantly different from one-twelfth will erode our belief in the symmetry of the die. But this is equally good evidence that we are sure of the link between probability and symmetry. In any case, the following more general claim seems correct: our subjectiveprobability that the physical probability of a red face is 1/12 is never lower than our subjectiveprobability that the die is symmetrical.

3. Rational probabilities can mean rational subjective probabilities (as it does forthe objectivist Bayesian) or logical probabilities, in the sense of Keynes (1921) and Carnap(1950).

4. There may be cases of non-enumerative statistical induction other than those in whichwe infer probabilities from symmetries. But I know of none.

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problem extends beyond the gaming halls. The second part considers the

principle of indifference in more detail, the third an alternative account of due to Henri Poincar and Hans Reichenbach. The nal part presents my

own account of .

1. The Success of Non-Enumerative Statistical Induction

It might be thought that successful is conned to a few unimportantcases. In what follows, I show that is widespread and that it has been andis essential to the sciences.

Gambling Devices: Perhaps the most conspicuous success of thoughnot the most importantis its use to infer the values of the probabilitiesassociated with gambling setups, such as the tossed coin, the thrown die, theroulette wheel, and the urn full of colored balls. Without any more empiricalinformation than that concerning the evident symmetries of these devices, wecan infer exact probabilities for various outcomes. (For example, we infer aone-half probability for a tossed coins landing heads, a three fths probability for drawing a red ball from an urn containing fty balls, thirty of which arered, and so on.)

Furthermore, as noted above, if we are convinced that our informationabout the symmetry is correct (i.e., that there is not some hidden asymmetry,as in a loaded die), then we will cling to our inference about the probability even given vast amounts of unfavorable enumerative evidence. For example,if I am sure that a coin is perfectly balanced and tossed fairly, no run of headswill be long enough to force me to abandon my belief that the probability of heads is one-half (see note 2).

Statistical Mechanics: The nineteenth century development of statistical

mechanics relied on certain probabilistic assumptions about the movementsof molecules, in particular, the assumption that in a given interval of time,all relevantly similar molecules are equally likely to be involved in a collision(rather as though the molecules that are to collide are determined by making

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selections from an urn). These assumptions, which allowed the derivation of

the ideal gas law, various other parts of the kinetic theory of gases, and thesecond law of thermodynamics, were obviously not based on enumerativeinduction. No one wasable to tabulate the frequencies of molecular collisions;on the contrary, the acceptance of the atomic theory of matter was largely brought about by the success of statistical mechanics. The probabilisticassumptions that provided the foundation for early statistical mechanicscan only have been inferred by . (For a historical sketch of the role of such assumptions in early statistical mechanics, see Ehrenfest and Ehrenfest(1959).)

The foundation (though not necessarily the truth) of these inferences hassince been questioned (Sklar 1993). But although physicists and philosophershave done their best to nd other foundations for our knowledge of theprobabilities of statistical mechanics, the fact remains that the distributionof those probabilities was rst successfully inferred from symmetries. Nofurther work on the foundations of statistical mechanics can erase this fact;the best the skeptic can do is to claim that the inference was a lucky accident.

Darwinian Explanation: Darwins tness is a probabilistic concept, mea-suring (roughly) the likelihood that an organism will survive and proliferate.(Thus survival of the ttest is not guaranteed: an unt but lucky organismmay be the one whose offspring inherit the earth. But it is overwhelmingly probable that the tter organism will bear this responsibility. See Mills andBeatty 1979.)

Darwinian explanation of the predominance of some trait usually takesas a premise that that trait confers or conferred greater tness than thealternatives.5 This premise is a premise about relative values of probabilities.Because the species that bore the traits in question may be long extinct,

5. Sometimes the trait itself does not increase tness, but is correlated with some othertness-increasing trait.

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enumerative tests of these probabilistic premises are not usually possible. The

probabilities (or at least, their relative magnitudes) must be inferred by way of some kind of non-enumerative induction.

An example will show that thisnon-enumerative inference of probabilitiesis at least sometimes based on symmetries. Consider the claim that the tnessof large-beaked Galpagos nches increases (relative to that of other nches)as the number of large seeds increases (relative to smaller seeds). The reasonfor the relative increase in tness is this: it is better for a small-beaked nchto nd a small seed, because the small beak cannot crack a large seed, but it isbetter for a large-beaked nch to nd a large seed, because large nches needthe extra nutrition to maintain their large bodies. As the proportion of largeseeds increases, the chance that the next seed a nch encounters will be largeincreases. Thus the probability of nding a favorable seed increases for largenches and decreases for small nches.

The argument for the claim about nch tness, then, is based on a claimabout probability: the more large seeds, the greater the probability of ndingone. It is obvious to anyone that this claim is true. The reasoning involvedis very similar to that concerning an urn full of balls (the urn is the island,

the balls, seeds). As such, it is an instance of inferring probabilities fromsymmetries. Incidentally, in the case described, years of careful eldwork haveprovided enumerative evidence that supports the non-enumerative inferencemade in a few seconds by the reader of this paper ( Weiner 1994).

The successes just described strongly suggest that there is some robustprocedure by which we go from symmetries to probabilities, and thus tofrequencies, with considerable success. But what is that procedure, and why does it work?

2. Inference From Epistemic Symmetries

The principle of indifference may be seen as the consequence of putting thefollowing constraint on the assignment of probabilities to a set of mutually

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exclusive outcomes:

Symmetries in the probability distribution ought to mirror symmetriesin our knowledge concerning the outcomes.

The principle is not much help when we have an abundance of relevant infor-mation, such as a mass of statistics, for then there are no useful symmetriesin our knowledge space. The less we know, the more the rule comes intoits own. When we have no information to distinguish two outcomes, it takesthe familiar form: assign two possibilities concerning which we are equally ignorant, equal probabilities.

There is a standard objection to the principle of indifference. The princi-ple, it is claimed, cannot be phrased so as to pick out a unique probability distribution. The argument is usually presented as entailing the inconsistency of the principle, as follows: because many different probability distributionssatisfy the principle, it simultaneously recommends the ascription of con-icting probabilities. Thus it cannot be taken seriously as an inferentialrule.

A well-known example that generates an inconsistent ascription of prob-

ability is that of the cube factory.6

The factory produces cubes with edgelength of up to two centimeters. What is the probability that the next cubeproduced has edge length of less than one centimeter? An answer: we know nothing about the edge length, except that it is less than 2 cm; thus, ourprobability distribution over the edge length should be distributed evenly over the interval between zero and two. It follows that the chance of the nextcube having side length of less than one is one-half. Another, inconsistent,answer: we know nothing about the cubes volume, except that it is less than8 cc; thus, our probability distribution over the cubes volume should bedistributed evenly over the interval between zero and eight. It follows that

6. The example is taken from van Fraassen (1989). It is based on the work of JosephBertrand (1889).

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the chance of the next cube having a volume of less than onethat is, the

chance that it will have a side length less than oneis one eighth.It is possible to fend off arguments of this sort by imposing an extra

constraint on the probability distribution, that it be invariant under certaintransformations. A case can be made for this requirement by appealing tosymmetries, which is certainly in the spirit of the original principleindeed,in recent years, something of a philosophical arms race has developed inthis area. The proponents of indifference have suggested further rules fordetermining relevant symmetries, while their opponents have offered new counterexamples to those rules. (For a good summary, with references, see van Fraassen 1989, chap. 12.)

The entire dispute is, I believe, irrelevant to the problem posed at thebeginning of this paper. The argument about the cube factory concerns theexistence of a method for picking out unique probability distributions, and(less explicitly) the rationality of such a method, should it exist. In the caseof the die and the other examples described in the last section, these issuessimply do not arise.

First, we already know that, in the cases I have described, it is possible to

arrive at a unique probability distribution. Upon viewing the dodecahedraldie for the rst time, we immediately see that the probability of any facelanding uppermost is one in twelve. The problem is not whether this can bedone, but how we do it.

Second, the question of the rationality of the one in twelve probability isquite secondary. What is striking is that the probability we choose is correct .It is this fact that must be explained. By contrast, there is no such fact tobe explained in the cube factory scenario, since we do not know the correctprobability distribution over cubes.7 (To gain such knowledge we would need

7. I take it that this claim is uncontroversial. Even proponents of the principle of indif-ference do not think that the principle gives us the correct physical probability distributionover cubes (see note 1).

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to look inside the factory, at the very least.)

In short, there are two quite distinct problems. The rst, exemplied by the case of the cube factory, is the problem of what probabilities, if any, can berationally ascribed to events we know nothing about. (The most interestingconsequence of solving this problem would be the justication of certainprior probability distributions for Bayesian inference; see Jeffreys (1939),Rosenkrantz (1981), and Jaynes (1983).) The second problem, exempliedby the case of the dodecahedral die, is the problem of how we successfully infer physical probabilities from symmetries. I have argued above that thisinference must be based on knowledge, not ignorance. Physical probabilitiesare inferred from physical symmetries, that is, symmetries that are not in themind but in the world.

3. Inference From Supercial Physical Symmetries

A proposal due to Henri Poincar (1905) and adopted in a modied versionby Hans Reichenbach (1949) founds on physical symmetries. This pro-posal is similar in certain ways to my own, but it is inadequate, I will argue,because it takes for granted, without explaining, an important inference fromsymmetry to probability.

Poincars account of is clearest in his treatment of the roulette wheel.The result of a game on the wheel, Poincar assumes, is determined by thewheels nal position. 8 The position is represented by a real-valued variableh ranging between 0 and 2p . Whether the wheel produces a red or a black number, then, is determined by h . Call values of h that correspond to redand black numbers respectively red and black values. Poincar notes that,in virtue of the physical symmetries of the roulette wheel (or more exactly,

of the way it is painted), red and black values of h form intervals of equal

8. Not all of Poincars comments are true of real roulette wheels. It is perhaps best tointerpret him as describing a simplied roulette wheel, in which there is a xed pointerinstead of a ball. When the wheel comes to rest, the number next to the pointer wins.

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size that alternate rapidly as h goes from 0 to 2p . (See gure 2, in which the

shaded regions represent red values, the white regions black values, of h .)From this symmetrythat is, from the physical symmetry of the pattern onthe wheelPoincar claims we may infer a one-half probability that a rednumber will be produced.

Figure 2: The probability of a red number is equal to the proportion of thearea under q(h ) that is shaded. After Reichenbach (1949).

His reasoning is as follows. Let q(h ) be the physical probability dis-tribution over h , which is unknown to us. The probability of obtaining ared number will be given by summing the probabilities that h falls into any particular red interval. More formally,

p(red) = 2p 0 f (h )q(h ) d h where f (h ) is equal to one for red values of h , and zero otherwise.

Now, provided that

1. The red and black intervals of h are quite small, and so alternate rapidly,and

2. q(h ) is fairly smooth, in the sense that it does not uctuate rapidly,

the above expression for the probability of red will be approximately equal tothe ratio of red to black intervals, that is, one-half. (More precisely, the ratiois one to one, which corresponds to a probability of one-half.) This resultholds because (a) a smooth q(h ) will remain approximately constant over

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any two neighboring (red and black) intervals, and (b) the ratio of any two

such neighbors is the same (one-half). We may be quite ignorant about thedetails of q(h ) , then, but provided we know that it is fairly smooth, we canproceed from the symmetry of the roulette wheels color scheme to a reliableestimate of the probability of a particular color.

The usual objection to symmetry-based approaches, as applied to thePoincar-Reichenbach approach, would be phrased as follows: why chooseh as the parameter with which to calculate the proportion of red to black outcomes, when there are many other parameters that would deliver differentsymmetries, and thus different probabilities? Poincar and Reichenbach havea ready answer at their disposal: we choose h because it is q(h ) that we know to be relatively smooth. (Thus the method for is as follows: nd a h forwhich q(h ) is smooth, and let the ratio of red to black in f (h ) dictate theprobability.) 9

The difcult question for Poincar and Reichenbach concerns our knowl-edge of q(h ) . How do we reach the conclusion that q(h ) is relatively smooth?The intuitive answer is that we expect q(h ) to be smooth because the roulettewheel has perfect circular symmetry. In fact, we do far better than this: from

the symmetry of the wheel, we infer that the probability distribution overh is uniform , that is, that the probability is the same that any h will be thewheels nal resting place. (To see that we are in fact inclined to reason in thisway, consider what happens when the symmetry is broken. If the wheel wereredesigned with little braking pads under the red sections, our estimate of q(h ) would no longer be uniform, or even, in the relevant sense, smooth.)

Thus our knowledge of q(h ) is itself grounded in the symmetry of theroulette wheel. It seems that the wheel is not the only thing that is goinground in circles. To make progress, Poincar and Reichenbach must explainthe basis of the inference that q(h ) is smooth.

9. Note that if h is a parameter for which red and black intervals are not equal, thenq(h ) and q(h ) cannot both be fairly smooth in the required sense.

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Poincar attempts to avoid this task by pointing out that, as the red and

black intervals become narrower and narrower and so alternate more andmore rapidly, q(h ) needs to be less and less smooth. In the limit, he argues, itis enough to suppose that q(h ) is continuous, which is, he contends, surely areasonable assumption. But in the limit here means as the number of redand black sections on the wheel increases to innity. Thus the assumptionof continuity is relevant only to a wheel divided into innitely many red andblack sections. As Reichenbach sees, that is not the problem we face, androutinely solve, in real life. There are thirty-six red and black sections on the(simplied) wheel, no more.

Reichenbach does not attempt to solve the problem. He simply points outthat the application of to the wheel depends on an assumption about thesmoothness of q(h ) . He thus explicitly neglects, as Poincar has implicitly neglected, the point at which truly operates: our inference from thesymmetry of the wheel to the smoothness of q(h ) .10

4. Inference From Physical Symmetries in the Mechanism

It is my claim that is based on symmetries in the mechanism of thechance setup in question. This position is, I think, intuitively appealing; inthis section I will show that it is also correct.11

10. It is worth pointing out a further problem with the Poincar-Reichenbach proposal,closely related to that just described. The proposal depends on there being two probability distributions that each describe the outcome of the same trial, one of which is continuous(for the wheel, h ). In many important cases, there is no such continuous variable. Whatcontinuous variable, for example, describes the outcome of a coin toss, or a drawing from anurn? Or a spin of a non-simplied roulette wheel in which each colored section has a hollow to capture the ball?

11. The approach taken in this section is similar in some respects to the method of

arbitrary functions that emerged from Poincars work, especially as developed by EberhardHopf (Hopf 1936; for summaries in English see von Plato (1983) and Engel (1992)).Hopf examines, as I do, the behavior of the function mapping initial conditions onto

outcomes, and shows that for some gambling devices, Poincars insight can be applied.However, like Poincar, Hopf is interested only in the behavior of these functions as certainparameters (the number of red and black sections or the initial velocity of the wheel)

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The rst section shows why, physically, there is a correlation between

certain symmetries and certain probabilities. The next two sections show how this physical fact serves as the basis for an account of .

4.1 Roulette Once More

Consider the simplied roulette wheel. Assume for now that the laws of nature are deterministic, at least in the form in which they apply to thewheel. (I will later show that this assumption is not necessary.) It followsthat the outcome of any game on the wheel is fully determined by some set

of initial conditions. In any chance setup there will be several aspects of these conditions that change from game to game; I will call these aspectsthe IC -variables . In the case of the wheel the sole -variable is the speedwith which the wheel is initially spun; call it x . (Most setups will have more

-variables; however, I will not deal with the multivariable case in this paper.There are no serious complications (as shown in Strevens (1996)).)

Suppose, then, that there is just one -variable x . Let e designate theevent of the wheels producing a red number. As a consequence of theassumption of determinism, the laws of nature together with the mechanismof the wheel determine a function he(x ) which is equal to one just in case xcauses a red number to be obtained on the wheel, zero otherwise.

approach innity. Since these parameters do not, in fact, approach innity, the physicalsignicance of Hopfs work is at best unclear.

Eduardo Engel has written a monograph extending Hopfs work in many ways ( Engel1992). The chief mathematical technique is, as in Hopf, to let parameters approach innity.But Engel also arrives at some results concerning the rate of convergence on the probability asthe parameters increase (for certain kinds of initial condition distribution functions). Theseresults do have physical signicance. His section on coin tosses is a case in point: he showsthat, for realistic parameters and a sufciently smooth initial condition distribution function,the probability of heads will be very close to one-half. This work has a rather different avorfrom mine; there is much talk of convergence (a property of the sequences generated when aparameter goes to innity), but nowhere is there a discussion of what I call microconstancy (a property of real physical systems in which parameters are xed). Also, Hopf and Engel arenot concerned with the epistemological problem addressed here. They seek to explain why the physical probability of a red value is one-half, not how we come to know this fact.

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Observe that he(x ) has the same sort of symmetry as the function f (h )

introduced in the last section. Red and black values of x form rapidly alternating bands; neighboring bands are of approximately equal size. (Seegure 3. Note that in he(x ) , unlike f (h ) , the red and black intervals slowly narrow. But they do so at the same rate, so that the ratio of a red interval toa neighboring black interval remains one to one.) The rapid alternation of he(x ) is a consequence of the fact that we can always change the outcome of a spin of the wheel by spinning a little faster or a little slower. The equal sizeof the black and red bands is a consequence of the physical symmetry of thewheel, in particular, of the fact that at any point in any spin, the wheel takesapproximately equal time to rotate through a red segment as it does to rotatethrough a black segment. I will say that such an he(x ) is microconstant withratio 0.5 (see the appendix for a formal denition of microconstancy).

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h e

Figure 3: A portion of the function he(x ) (equal to one when x produces ared outcome, otherwise zero)

Despite the similarity between he(x ) and f (h ) , they are not to be con-fused. Whereas he(x ) expresses facts about the mechanism of the wheel, f (h )expresses facts only about its paint scheme.

Now consider the probability distribution q(x ) over the -variable x .(What I have to say is neutral with respect to the interpretation of q(x ) . My only assumption is that there is such a distribution. If nothing else, we can

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always take q(x ) to represent frequencies of actual values of x .)12 As with

he(x ) and f (h ) , there is an analogy between q(x ) and q(h ) , but it is notexact: a value of h describes an outcome of a game on the wheel, while a valueof x describes an initial condition of such a game.

The probability of the wheels producing a red number can be derivedfrom q(x ) and he(x ) . It is simply the total probability of the red valuesof x (i.e. those values of x that result in a red number). The function he(x )tells us which values of x are red; the density q(x ) tells us how likely these values are. If e is the event of obtaining a red number,

p( red) = b

ahe(x )q(x ) d x

where a and b are the minimum and maximum possible values of x .We can now apply Poincars purely mathematical observation to he(x )

and q(x ) . For any sufciently smooth q(x ) , the probability of red isbecause of the geometry of he(x )approximately equal to one-half. Thus,provided that we know that q(x ) is reasonably smooth, we need know noth-ing more to infer a probability for e of one-half. We can use the micro-constancy of he(x ) to parlay some very general (probabilistic) informationabout the initial condition x into an exact value for the probability of a rednumber .13

The same sort of treatment can be applied to any of the other familiargambling devices (the die, the urn full of balls, a tossed coin, etc.). In eachcase, due to the symmetry of the apparatus, the function he(x ) is micro-constant: it characterized by thin, evenly proportioned stripes (provided, of

12. A frequency interpretation of the initial condition distribution function introduces atechnical difculty dealt with in Strevens (1996).

13. If you are inclined to doubt that croupiers functions q(x ) are smooth, consider this:if the functions are not smooth in the relevant sense, it would very likely be the case thatdifferent croupiers would turn up red numbers with different frequencies. Casinos havespent much time and money ensuring that this is not the case; thus, for gambling devices atleast, it is reasonable to suppose that the relevant functions q(x ) are smooth. See the nextsection for further discussion.

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course, that e is one of the basic outcomes, such as a six being thrown or a

particular ball being drawn). On the assumption that the relevant q(x ) issmooth, we can infer that the probability of the event e is 1/n, where n is thenumber of basic outcomes. (A more formal, more general treatment is givenin the appendix.)

This result provides a basis for , as follows. Given a chance setup forwhich

1. We have reason to think that the initial conditions are distributedsmoothly, and

2. We can infer from the symmetries of the setup mechanism that he(x )is microconstant, with constant ratio p,

we can infer that the probability of an e event is p.What is the role of the symmetries in such an inference? The symmetries

do not themselves tell us the values of probabilities. Rather, they allow us toconvert a very vague, very general piece of knowledge about probabilitiesthat x is smoothly distributedinto a very specic piece of information, anexact value for the probability of an e event.

In the next two sections I consider the circumstances under which (a)and (b) above hold true, that is, the circumstances in which successful ispossible.

4.2 Knowledge of the Distribution of Initial Conditions

Non-enumerative statistical induction or requires a probabilistic premise,namely, that the probability distribution over the relevant initial conditionsis smooth. Where could this information come from? How can we know that some initial condition distribution function q(x ) is smooth? How canwe come to have knowledge of the probabilistic distribution of an -variablesuch as the speed with which a roulette wheel is spun? Knowledge of q(x )

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must be supplied either by enumerative induction or by non-enumerative

induction .14

In the case of the roulette wheel, some kind of enumerative induction isprobably at work. We simply know from experience that human actions suchas the twirl of a wheel produce results that are smoothly heaped around someaverage value. (The experience is often gained while trying to produce resultsthat require a far more nely honed distribution.)

We might also be able to arrive at a non-enumerative inference to theeffect that q(x ) is smooth, if we knew enough about human physiology thatwe had some idea about the causes of variation in spins. This inference woulddepend on the distribution of some set of physiological initial conditions,which would have to be inferred in turn. At some point some enumerativeinduction must be done to get non-enumerative induction off the ground.

This initial enumerative induction might be on a grand scale. Call thekinds of variables in terms of which we usually work our standard vari-ables.15 It seems to be the case that, for whatever reason, our standard vari-ables are usually smoothly distributed. If we go ahead and generalize fromthis observation (by enumerative induction), we arrive at the conclusion that

most standard variable distributions are smooth. We may consequently takeourselves to have empirical grounds for adopting a revised and differently deployed principle of insufcient reason of the following form:

In theabsence of any reason to think otherwise, assume that any standard variable is fairly smoothly distributed.

14. Leonard Savage (1954) has suggested, in connection with Poincars explanation of , that a distribution suchas q(x ) canbe interpreted as a subjective probabilitydistribution.

But while this might explain the fact that we agree on a probability for red (because we allhave a smooth subjective probability distribution over x ), it cannot explain the objectivecorrectness of the agreed-on answer (nor did Savage mean it to do so).

15. One way of characterizing standard variables is as follows: they are those variablesthat induce measures directly proportional to the units. Thus inches are standard measuresof length, while inches* are not, where an objects length x in inches* is derived from itslength x in inches as follows: x = sin x + x .

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Let me stress that I am not proposing that our standard variables have any

special logical status. They are simply the variables with which we preferto operate, and which are, conveniently for us, for the most part smoothly distributed.

Note that this principle of insufcient reason is defeasible. Sometimeswe have sufcient reason to think that a particular standard variable h isnot smoothly distributed. Consider a h determined by a mechanism that,given a smoothly distributed q(x ) as input, produces a ruggedly distributedprobability distribution p(h ) over its output. An example is my roulettewheel with brakes. We clearly ought not to assume that such a h is smooth.

4.3 Knowledge of Microconstancy

The second premise required for is that the relevant he(x ) is micro-constant with some constant ratio p. Such information must come fromknowledge of the probabilistic mechanism in question together with knowl-edge of the laws of nature. There arises the issue of exactly which aspects of the mechanism and laws are relevant, a question made more interestingby thefact that we seem to get by with a small portion of the complete information.(Physics failures may be gambling adepts.)

What is necessary is whatever knowledge guarantees the existence of relevant symmetries in the operation of the mechanism, for example, whateverentails that a spinning coin takes about the same time for each half-revolution,or that a spinning roulette wheel takes about the same time for each 1/36th of a revolution. Such knowledge comes in two complementary parts: (a) whatthe laws of motion care about (distribution of mass, brake pads), and (b)what the laws do not care about (the color of a segment of the wheel, or the

image on one side of a coin). The use of these facts to infer microconstancy does not require us to attend to the exact form of the dynamical laws of nature.Indeed, the facts remain the same (for the most familiar mechanisms at least)whether the laws of nature are Aristotelian, Newtonian, those of medieval

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impetus theory, or those of quantum mechanics. Thus the non-physicists

evident skill at pinpointing certain probabilities. (This observation has afurther consequence: insofar as the laws of, say, evolutionary biology arebased on probabilities founded in microconstancy, those laws do not dependon the details of the laws of physics, but only certain very general symmetriesof the laws. See Strevens (1996) for the implications of microconstancy forreduction.)

That microconstancy is found in common gambling devices is reasonably clear. What, however, of ecosystems? What mechanism do we examine inorder to divine a probability, and what symmetries do we look for? I haveanswered these questions in some detail elsewhere ( Strevens 1996). Themethod, very roughly, is to regard the entire ecosystem as the mechanism,and then to ask which things affect, and which things fail to affect, themovement of the mechanisms partsthe falling seeds, the foraging nches,and so on. Exact probabilities may not be forthcoming, but qualitative results(concerning, for example, the relative advantages of large and small beaks)are within reach.

Finally, how can we apply this method of to chance setups involving

indeterministic mechanisms? In such cases, the effects of the indeterminismcan be represented by an extra -variable w , representing the error causedby quantum uctuations. This leaves us with a function mapping initialconditions to outcomes that is, in effect, deterministic. We may examinethe symmetries of the mechanism as before, and, provided that q(w ) issufciently smooth, infer a corresponding probability.

5. Conclusion

Due to the microconstancy of certain mechanisms, and the smoothness of the distributions of certain variables, there is a correlation, in our world,between symmetry and probability, and thus (in a world where frequenciesusually reect probabilities) between symmetry and frequency.

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Our species has evidently picked up on this correlation, whether through

learning or natural selection, and has exploited it for some time. Yet thereason for the correlation has not been clearly understood. It is because of this lack of understanding, I suggest, that the principle of indifference hasbeen able to exert its hold. In what became the paradigmatic probabilis-tic experimentsthose involving gambling devices such as the tossed coinand diethere is a direct correspondence between the symmetries in themechanism and the symmetries in our knowledge about outcomes. Previ-ous thinkers correctly saw that symmetries lead to reasonable beliefs aboutprobability, but in framing the rule that enshrines this sort of inference, they wrongly took the epistemic symmetries, not the physical symmetries, as theproper source of these beliefs. In those cases where the beliefs turned out tobe true, it appeared that provided a priori knowledge of probabilities.

The truth about is not as glamorous as that. Probabilities are notconjured from nothing, but are introduced in the premises of the inferencein the form of an assumption about the distribution of initial conditions.Furthermore, empirical knowledge plays, if anything, a greater role in careful

than in enumerative statistical induction, for is founded not just on

knowledge of initial condition distributions, but also on fallible, empiricalknowledge of the physical constitution of the relevant mechanism and of thepertinent laws of nature.

Yet knowledge of probabilities inferred from symmetries is knowledgeacquired with little or no experiment in the traditional sense. It may not bea priori, but it is not experimental, either. As such, the further philosoph-ical and psychological study of has the potential to cast light on thosesciencessuch as statistical mechanics, evolutionary biology, and certain varieties of social sciencein which experiment has played only a subsidiary role.

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Appendix

Call a function he(f ) indicating which values of an -variable f producea given outcome e, as described in the main text, an evolution function. Iprovide in this appendix a more precise characterization of the mathematicalproperties of evolution functions that make possible. For further details,as well as a discussion of a realistic case (which requires a method for dealingwith multiple -variables), see Strevens (1996).

The strike ratio of an evolution function he(f ) over an interval [x , y ] of f is the proportion of the interval for which he(f ) is equal to 1.

An evolution function he(f ) sconstant ratio index is determined as follows.Consider a partition of the range of f into intervals [x , y ]. Call such a partitiona constant ratio partition if the strike ratio for each [x , y ] is more or less thesame. Call the size of the largest interval in such a partition the partitionsconstant ratio index ( ). Take the constant ratio partition of f that yieldsthe smallest q. That is the evolution function's constant ratio index. (If he(f ) is constant, its is zero.) If an evolution function has a of q, thenany segment of he( f ) much longer than q will have the same strike ratio. Insuch a case it makes sense to talk of the strike ratio of the entire evolutionfunction.

An evolution function is microconstant if it has a that is very small.Theorem: If an evolution function is microconstant, with strike ratio

equal to p, then for any q(f ) that is relatively smooth (that is, for any q(f )nearly constant over most small intervals of f ), the probability of e is approxi-mately equal to p.

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